Final Exam December 13, 2016

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Marilynn Wilkins
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1 Final Exam Instructions: You have 120 minutes to complete this exam. This is a closed-boo, closed-notes exam. You are allowed to use a calculator during the exam. Do NOT unstaple your exam. Do NOT write on the bacside of any page. Wor included on the bac of a page will NOT be graded. All wor must follow a logical thought process. Otherwise, the wor presented will NOT receive full credit. Coordinate axes must be included in your solution, if they are not already provided in the problem statement. Appropriate free body diagrams must be included. Usage of mobile phones and other electronic communication devices is NOT permitted during the exam. Problem 1 (20 pts) 2 (24 pts) 3 (20 pts) 4 (36 pts) Total Score Name (please print): Address: Instructor (please circle): Rhoads Arrieta Final Exam Page 1 of 25

The dis is released from rest with its center of mass G located a distance e from the dis s geometric center O, and

2 Problem 1 (20 points): Given: An inhomogeneous dis (of mass m, radius R, and with a centroidal radius of gyration G ) sits upon a smooth horizontal surface. The dis is released from rest with its center of mass G located a distance e from the dis s geometric center O, and oriented an angle θ as measured from the vertical. Find: Determine the angular acceleration of the dis upon release. Express your answer in terms of no more than: g, e, G, R, and θ. Final Exam Page 2 of 25

3 This page is for extra wor related to Problem 1. Final Exam Page 3 of 25

4 This page is for extra wor related to Problem 1. Final Exam Page 4 of 25

5 Problem 2 (24 points): Given: The system shown below consists of a pulley (of mass m and centroidal mass moment of inertia I O ) and bloc A (of mass 2m). The pulley is actuated by an applied moment given by M(t). The cable between the pulley and bloc remains in tension while the system is in motion. Find: (a) Draw the free body diagram(s) necessary to derive the equation of motion for the system; (b) Derive the equation of motion for the system in terms of the coordinate θ; (c) Find the system s natural frequency ω n ; (d) Find the particular solution θ p (t) associated with the equation of motion if M(t) = M 0 sin ωt and gravity is negligible; and (e) Determine if the particular solution is in-phase or out-of-phase with the excitation if ω = ω n /2. Final Exam Page 5 of 25

6 This page is for extra wor related to Problem 2. Final Exam Page 6 of 25

7 This page is for extra wor related to Problem 2. Final Exam Page 7 of 25

8 Problem 3 (20 points): Consider the mass-spring-dashpot system (left) and corresponding free body diagrams (right) shown below. m A g F 1 A F 2 A B N A m B g F 3 x A x B B F 4 F 5 N B Problem 3.A: The spring force F 5 is given by (note the direction of the force indicated in the free body diagram): TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA (a) F sp = x B (b) F sp = x B (c) F sp = (x A + x B ) (d) F sp = (x A + x B ) Problem 3.B: The spring force F 3 is given by (note the direction of the force indicated in the free body diagram): (a) F sp = x B (b) F sp = x B (c) F sp = (x A + x B ) (d) F sp = (x A x B ) Final Exam Page 8 of 25

Consider the mass-spring-dashpot system (left) and corresponding free body diagrams (right) shown below. m A g F 1 A F 2 A B N A m B g F 3 x A x B B F 4 F 5 N B Problem 3.

9 Consider the mass-spring-dashpot system (left) and corresponding free body diagrams (right) shown below. m A g F 1 A F 2 A B N A m B g F 3 x A x B B F 4 F 5 N B Problem 3.C: The spring force F 2 is given by (note the direction of the force indicated in the free body diagram): TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA (a) F sp = x B (b) F sp = x B (c) F sp = (x A x B ) (d) F sp = (x B x A ) Problem 3.D: The spring force F 4 is given by (note the direction of the force indicated in the free body diagram): (a) F sp = x B (b) F sp = x B (c) F sp = (x A x B ) (d) F sp = (x B x A ) Final Exam Page 9 of 25

10 The following equation of motion has been derived for a single-degree-of-freedom system: 3ẍ + 36ẋ + 300x = 0, where x is given in meters. Problem 3.E: The undamped natural frequency ω n of this system is: (a) ω n = 10 Hz (b) ω n = 10 rad/s (c) ω n = 300 Hz (d) ω n = 300 rad/s Problem 3.F: The damping ratio ζ of this system is: (a) ζ = 0.2 (b) ζ = 0.4 (c) ζ = 0.6 (d) ζ = 1.2 Final Exam Page 10 of 25

11 The following equation of motion has been derived for a single-degree-of-freedom system: 3ẍ + 36ẋ + 300x = 0, where x is given in meters. Problem 3.G: The free response of this system is best described as: (a) Undamped (b) Underdamped (c) Critically damped (d) Overdamped Problem 3.H: The damped natural frequency associated with this system is: (a) ω d = 4 Hz (b) ω d = 4 rad/s (c) ω d = 8 Hz (d) ω d = 8 rad/s Final Exam Page 11 of 25

12 Problem 3.I: In an undamped, mass-spring system, the forced response is maximized when the excitation frequency is identical to the natural frequency of the system. (a) True (b) False Problem 3.J: In a damped, mass-spring-dashpot system, the forced response is maximized when the excitation frequency is identical to the natural frequency of the system. (a) True (b) False Final Exam Page 12 of 25

Final Exam April 30, 2013

Final Exam April 30, 2013 Final Exam Instructions: You have 120 minutes to complete this exam. This is a closed-book, closed-notes exam. You are allowed to use a calculator during the exam. Usage of mobile phones and other electronic